1. Addition Subtraction Multiplication Division
Add Subtract Multiply Divide
Added to Subtracted from Multiplied by Divided by
Sum Difference Product Quotient
Increased by Decreased by Times
Diminished by Twice (two times)
(For other words or phrases see your textbook.)
2. 1st: Any operation inside a symbol of grouping or above or below a fraction bar
2nd: Exponentiation
3rd: Multiplication and division (left to right)
4th: Addition and subtraction (left to right)
3. Phrase Expression
a. The sum of two and x divided by three times y (2 + x)/3y
b. Five divided by x decreased by two 5/x - 2
c. Twice x divided by the sum of three and z 2x/(3 + z)
d. Three times the sum of x and 2 3(x + 2)
4. Sentence Equation
a. If twice a number is increased by 2 2x + 2 = 14
the sum is fourteen. (x denotes the number)
b. If the quotient of y divided by 3 is y/3 + 2 = 7
increased by 2 the sum is 7.
c. The sum of twice an integer and the integer 2x + x = x + 14
itself is equal to the integer plus 14. (x denotes the integer)
d. If three is subtracted from the sum of y and z (y + z) - 3 = 12
the result is 12.
5. a. Let x denote Mary's age then 3x denotes Tom's age.
3x + x = 52 (Combine like terms 3x + x or 3x + 1x = 4x)
4x = 52 (Divide each side by 4)
4x/4 = 52/4 (Simplify each side)
x = 13
Conclusion: Mary is thirteen years old.
b. Let x denote the amount of each payment.
Downpayment + Balance Due = Total Amount Owed
150 + 12x = 1350 (Subtract 150 from each side)
150 + 12x - 150 = 1350 - 150 (Simplify each side)
12x = 1200 (Divide each side by 12)
12x/12 = 1200/12 (Simplify)
x = 100
Conclusion: The amount of each payment is $100.
c. Let x denote the temperature before the front hit.
x - 15 = 13 (Add 15 to each side)
x - 15 + 15 = 13 + 15 (Simplify each side)
x = 28
Conclusion: The temperature was 28 degrees before the front hit.
d. Let x denote one number then the other is 2x + 6 denotes the other number.
x + (2x + 6) = 69 (Remove parentheses)
x + 2x + 6 = 69 (Combine like terms)
3x + 6 = 69 (Subtract 6 from each side)
3x + 6 - 6 = 69 - 6 (Simplify each side)
3x = 63 (Divide each side by 3)
3x/3 = 63/3 (Simplify each side)
x = 21
Conclusion: One number is 21, the other is 2x + 6 = 2(21) + 6 = 42 + 6 = 48.
6. a. -16/8 - 4 (Divide before you subtract)
= -2 - 4 (Subtract, i.e. add the opposite)
= -6
b. [12 - (-3)(-2)]/[-7 - 3(-1) + 1] (Perform operations above and below
= [12 - 6]/[-7-(-3)+1] fraction bar first. Above fraction bar
= [6]/[-4+1] there are two operations. Multiply before
= 6/[-3] you subtract. Below fraction bar there are
= -2 three operations. Multiply before you add
or subtract. With addition and subtraction
work from left to right. Lastly do the division
that is indicated.)
c. 6 - (2 + [ 5 - (-4) ] + (-7) ) (With nested symbols of grouping perform the
= 6 - (2 + [9] + (-7) ) operation inside the intermost symbol of grouping
= 6 - (11 + (-7) ) first, i.e. subtract -4 from 5. Next do the
= 6 - 4 addition operations inside the parenthesis.
= 2 Lastly subtract.)
d. 12/6 - 2(5) (With the three operations of division, subtraction,
= 2 - 10 and multiplication you always multiply and divide before
= -8 you subtract. Multiplication and division are always
performed from left to right, hence, in this case you
divide before you multiply. After dividing and
multiplying you subtract.
e. 3/4 + 2/6 (To add fractions you must have a common denominatior.
=9/12 + 4/12 The LCD is 12. Express each as equaivalent fractions
=13/12 with denominator 12. To add, add the numerators and
divide by the common denominator.)
f. -2/5 - 1/3 (To subtract fractions you must have a common denominator.
= -6/15 - 5/15 The LCD is 15. Write each fraction as an equivalent
= (-6 - 5)/15 fraction with denominator 15. To subtract, subtract
= -11/15 numerator of the second, from numerator of the first and
divide by the common denominator.)
g. [-7/15][3/4] (To multiply fractions, divide numerators and
= [-7/5][1/4] denominators by any common factors, in this example
= -7/20 there is a common factor of 3. After recucing
multiply theri numerators together and divide by
the product of their denominators.)
h. Divide 4/9 by 5/3. (To divide one fraction by another, multiply by the
[4/9][3/5] reciprocal of the divisor. Remember to reduce before
= [4/3][1/5] you multiply. In this case we can reduce by dividing
= 4/15 numerator and denominator by 3.)
7. Solve for x:
a. x - 7 = -5 (Add 7 to each side.)
x - 7 + 7 = -5 + 7 (Simplify)
x = 2
b. -3x + 7 = 13 (Subtract 7 from each side.)
-3x + 7 - 7 = 13 - 7 (Simplify.)
-3x = 6 (Divide each side by -3.)
-3x/-3 = 6/-3 (Simplify.)
x = -2
c. 4x + 1 = -5 (Subtract 1 from each side.)
4x + 1 - 1 = -5 - 1 (Simplify.)
4x = -6 (Divide each side by 4.)
4x/4 = -6/4 (Simplify. Remember to reduce fraction to lowest term)
x = -3/2 .
d. x/2 - 3 = -8 (Add three to each side.)
x/2 - 3 + 3 = -8 + 3 (Simplify.)
x/2 = -5 (Multiply each side by 2.)
2(x/2) = 2(-5) (Simplify.)
x = -10
e. 2(x - 1) + 3x = 8 (Use distributive property to remove grouping.)
2x - 2 + 3x = 8 (Combine like terms.)
5x - 2 = 8 (Add two to each side.)
5x - 2 + 2 = 8 + 2 (Simplify.)
5x = 10 (Divide each side by 5.)
x = 2
8. Simplify.
a. 2x² - 3y + 7x² + 10y (Combine like terms.)
= 9x² + 7y
b. (3x - 5) - (-2x + 7) (Remove grouping.)
= 3x - 5 + 2x - 7 (Combine like terms.)
= 5x - 12
c. 5(3x - 2) + 4(x + 3) (Use distributive property.)
= 15X - 10 + 4X + 12 (Combine like terms.
= 19x + 2
9. Give an example of
a. A real number that is not rational.
The square root of 3.
b. A rational number that is not an integer.
2/3
c. An integer that is not a whole number.
-7
d. A whole number that is not a natural number.
0
1. Addition Subtraction Multiplication Division
Add Subtract Multiply Divide
Added to Subtracted from Multiplied by Divided by
Sum Difference Product Quotient
Increased by Decreased by Times
Diminished by Twice (two times)
(For other words or phrases see your textbook.)
2. 1st: Any operation inside a symbol of grouping or above or below a fraction bar
2nd: Exponentiation
3rd: Multiplication and division (left to right)
4th: Addition and subtraction (left to right)
3. Phrase Expression
a. The sum of two and x divided by three times y (2 + x)/3y
b. Five divided by x decreased by two 5/x - 2
c. Twice x divided by the sum of three and z 2x/(3 + z)
d. Three times the sum of x and 2 3(x + 2)
4. Sentence Equation
a. If twice a number is increased by 2 2x + 2 = 14
the sum is fourteen. (x denotes the number)
b. If the quotient of y divided by 3 is y/3 + 2 = 7
increased by 2 the sum is 7.
c. The sum of twice an integer and the integer 2x + x = x + 14
itself is equal to the integer plus 14. (x denotes the integer)
d. If three is subtracted from the sum of y and z (y + z) - 3 = 12
the result is 12.
5. a. Let x denote Mary's age then 3x denotes Tom's age.
3x + x = 52 (Combine like terms 3x + x or 3x + 1x = 4x)
4x = 52 (Divide each side by 4)
4x/4 = 52/4 (Simplify each side)
x = 13
Conclusion: Mary is thirteen years old.
b. Let x denote the amount of each payment.
Downpayment + Balance Due = Total Amount Owed
150 + 12x = 1350 (Subtract 150 from each side)
150 + 12x - 150 = 1350 - 150 (Simplify each side)
12x = 1200 (Divide each side by 12)
12x/12 = 1200/12 (Simplify)
x = 100
Conclusion: The amount of each payment is $100.
c. Let x denote the temperature before the front hit.
x - 15 = 13 (Add 15 to each side)
x - 15 + 15 = 13 + 15 (Simplify each side)
x = 28
Conclusion: The temperature was 28 degrees before the front hit.
d. Let x denote one number then the other is 2x + 6 denotes the other number.
x + (2x + 6) = 69 (Remove parentheses)
x + 2x + 6 = 69 (Combine like terms)
3x + 6 = 69 (Subtract 6 from each side)
3x + 6 - 6 = 69 - 6 (Simplify each side)
3x = 63 (Divide each side by 3)
3x/3 = 63/3 (Simplify each side)
x = 21
Conclusion: One number is 21, the other is 2x + 6 = 2(21) + 6 = 42 + 6 = 48.
6. a. -16/8 - 4 (Divide before you subtract)
= -2 - 4 (Subtract, i.e. add the opposite)
= -6
b. [12 - (-3)(-2)]/[-7 - 3(-1) + 1] (Perform operations above and below
= [12 - 6]/[-7-(-3)+1] fraction bar first. Above fraction bar
= [6]/[-4+1] there are two operations. Multiply before
= 6/[-3] you subtract. Below fraction bar there are
= -2 three operations. Multiply before you add
or subtract. With addition and subtraction
work from left to right. Lastly do the division
that is indicated.)
c. 6 - (2 + [ 5 - (-4) ] + (-7) ) (With nested symbols of grouping perform the
= 6 - (2 + [9] + (-7) ) operation inside the intermost symbol of grouping
= 6 - (11 + (-7) ) first, i.e. subtract -4 from 5. Next do the
= 6 - 4 addition operations inside the parenthesis.
= 2 Lastly subtract.)
d. 12/6 - 2(5) (With the three operations of division, subtraction,
= 2 - 10 and multiplication you always multiply and divide before
= -8 you subtract. Multiplication and division are always
performed from left to right, hence, in this case you
divide before you multiply. After dividing and
multiplying you subtract.
e. 3/4 + 2/6 (To add fractions you must have a common denominatior.
=9/12 + 4/12 The LCD is 12. Express each as equaivalent fractions
=13/12 with denominator 12. To add, add the numerators and
divide by the common denominator.)
f. -2/5 - 1/3 (To subtract fractions you must have a common denominator.
= -6/15 - 5/15 The LCD is 15. Write each fraction as an equivalent
= (-6 - 5)/15 fraction with denominator 15. To subtract, subtract
= -11/15 numerator of the second, from numerator of the first and
divide by the common denominator.)
g. [-7/15][3/4] (To multiply fractions, divide numerators and
= [-7/5][1/4] denominators by any common factors, in this example
= -7/20 there is a common factor of 3. After recucing
multiply theri numerators together and divide by
the product of their denominators.)
h. Divide 4/9 by 5/3. (To divide one fraction by another, multiply by the
[4/9][3/5] reciprocal of the divisor. Remember to reduce before
= [4/3][1/5] you multiply. In this case we can reduce by dividing
= 4/15 numerator and denominator by 3.)
7. Solve for x:
a. x - 7 = -5 (Add 7 to each side.)
x - 7 + 7 = -5 + 7 (Simplify)
x = 2
b. -3x + 7 = 13 (Subtract 7 from each side.)
-3x + 7 - 7 = 13 - 7 (Simplify.)
-3x = 6 (Divide each side by -3.)
-3x/-3 = 6/-3 (Simplify.)
x = -2
c. 4x + 1 = -5 (Subtract 1 from each side.)
4x + 1 - 1 = -5 - 1 (Simplify.)
4x = -6 (Divide each side by 4.)
4x/4 = -6/4 (Simplify. Remember to reduce fraction to lowest term)
x = -3/2 .
d. x/2 - 3 = -8 (Add three to each side.)
x/2 - 3 + 3 = -8 + 3 (Simplify.)
x/2 = -5 (Multiply each side by 2.)
2(x/2) = 2(-5) (Simplify.)
x = -10
e. 2(x - 1) + 3x = 8 (Use distributive property to remove grouping.)
2x - 2 + 3x = 8 (Combine like terms.)
5x - 2 = 8 (Add two to each side.)
5x - 2 + 2 = 8 + 2 (Simplify.)
5x = 10 (Divide each side by 5.)
x = 2
8. Simplify.
a. 2x² - 3y + 7x² + 10y (Combine like terms.)
= 9x² + 7y
b. (3x - 5) - (-2x + 7) (Remove grouping.)
= 3x - 5 + 2x - 7 (Combine like terms.)
= 5x - 12
c. 5(3x - 2) + 4(x + 3) (Use distributive property.)
= 15X - 10 + 4X + 12 (Combine like terms.
= 19x + 2
9. Give an example of
a. A real number that is not rational.
The square root of 3.
b. A rational number that is not an integer.
2/3
c. An integer that is not a whole number.
-7
d. A whole number that is not a natural number.
0